A -form on is called exact if for some form on and is closed if . Since for any form, as we proved in a previous lecture. The Poincaré lemma asserts that the converse is true:

Thm 1 (Poincaré Lemma) Let be a closed -form on with . Then is exact, i.e., there is a -form such that .

Stokes Theorem on parallelpipeds

In proving the Poincaré Lemma we will use Stokes theorem for integration over parallelpipeds in . Since this is technically different than the integration over oriented surfaces that we considered before, let us start by formulating the precise result here.

The unit -cube is the set . A parameterized -parallelpiped is the restriction to of a map where is an open neighborhood of in . We will use the notation to denote the parallelpiped as well as the map.

The boundary of the cube is covered by parameterized -parallelpipeds

and

for . However, these maps are not oriented consistently. The correct definition of the oriented boundary of a parallelpiped is suggested by the following theorem

Thm 2 (Stoke’s Theorem for parallelpipeds)Let be a parameterized -parallelpiped and let be a form defined on a neighborhood of . Then

where and .

To formulate the boundary of a parallelpiped we introduce the notion of a chain of parallelpipeds. A chain of – parallelpipeds is a formal sum

where and are -parallelpipeds. The summation and multiplication by integers are not the usual summation of maps, instead the notation is a formal sum intended to indicate that

for any -form . Based on \vpageref{eq:stokesparallelpiped} we define the oriented boundary of to be the following chain of -parallelpipeds

Proof:(Of Theorem 2) First suppose is a form defined on a neighborhood of in , where

Then

and

by the fundamental theorem of calculus. Since for we see that (1) holds for .

The identity now follows for a general form on by linearity and for form defined in a neighborhood of a -parallelpiped in by pulling the integrals back to .

In the proof of the Poincaré lemma we will use affine parallelpipeds. Given , we define the affine parallelpiped at with sides to be the parallelpiped given by the map

and denote the parallelpiped by

The oriented boundary of an affine parallelpiped is a chain of affine parallelpipeds:

where

and

Differentiation of forms

So far we have treated exterior differentiation of forms in a formal way. In the proof of the Poincaré Lemma we will need the following Lemma which shows that exterior differentiation can be expressed in terms of difference quotients:

Lemma 3Let be a -form defined on an open subset . Then

Proof: By linearity it suffices to consider where is an elementary -form and . Then

Proof of the Poincaré Lemma

To fix ideas, first consider the case . Suppose we are given a closed -form . We must construct a function such that . This is done by integrating :

where is the line segment from to (which is the affine -parallelpiped at with side ). Explicitly . Now consider the difference . The key observation is that, by Stoke’s Theorem,

Before we verify this identity, let us see how it implies that . Replacing by we see that

since . Since is continuous it now follows, on taking to , that

Thus the directional derivatives of exist and agree with . It follows that .

To prove (2), consider the triangle in bounded by the line segments , , . (Draw a picture!) This triangle can be expressed as the image of the following parameterized -parallelpiped,

The boundary of is the chain

Note that the boundary component is singular; it’s image is the single point and for any -form . Since we have, by Stoke’s Theorem 2,

The proof for is a direct generalization of the above. The triangle above is replaced by a cone over a parallelpiped, defined as follows. Let denote and affine -parallelpiped in . The cone over is the parameterized -parallelpiped given by the map

The boundary of is then the following chain

where . Since is degenerate as above, we put this succinctly as

where the cone over a chain of parallelpipeds is defined term by term.

Now let be a closed -form in , and let be the following form

Then for any affine parallelpiped we have

Thus, since , we have

for any affine -parallelpiped by Stoke’s theorem and the identity derived above.